They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications.
DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.
Indeed, the solutions of such problems may involve strong gradients (and even discontinuities) so that classical finite element methods fail, while finite volume methods are restricted to low order approximations.
Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations.
In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation.
The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually.
However, among the early influential contributors were Babuška, J.-L. Lions, Joachim Nitsche and Miloš Zlámal.
DG methods for elliptic problems were already developed in a paper by Garth Baker in the setting of 4th order equations in 1977.
A more complete account of the historical development and an introduction to DG methods for elliptic problems is given in a publication by Arnold, Brezzi, Cockburn and Marini.
A number of research directions and challenges on DG methods are collected in the proceedings volume edited by Cockburn, Karniadakis and Shu.
Unlike traditional CG methods that are conforming, the DG method works over a trial space of functions that are only piecewise continuous, and thus often comprise more inclusive function spaces than the finite-dimensional inner product subspaces utilized in conforming methods.
Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain
the solution is represented by Then similarly choosing a test function multiplying the continuity equation by
and integrating by parts in space, the semidiscrete DG formulation becomes: A scalar hyperbolic conservation law is of the form where one tries to solve for the unknown scalar function
-space will be discretized as Furthermore, we need the following definitions We derive the basis representation for the function space of our solution
denotes the space of polynomials of maximal degree
At first we make use of a specific polynomial basis on the interval
which fulfill the orthonormality relation Transformation onto an interval
Besides, prism bases are employed for planar-like structures, and are capable for 2-D/3-D hybridation.
The conservation law is transformed into its weak form by multiplying with test functions, and integration over test intervals By using partial integration one is left with The fluxes at the interfaces are approximated by numerical fluxes
: The interior penalty discontinuous Galerkin (IPDG) method is: find
for the symmetric interior penalty Galerkin method; it is equal to
In 2009, Liu and Yan first proposed the DDG method for solving diffusion equations.
[1][2] The advantage of this method compared with Discontinuous Galerkin method is that the direct discontinuous Galerkin method derives the numerical format by directly taking the numerical flux of the function and the first derivative term without introducing intermediate variables.
We can still get reasonable numerical results by using this method, and due to the simpler derivation process, the amount of calculation is greatly reduced.
It mainly includes transforming the problem into variational form, regional unit splitting, constructing basis functions, forming and solving discontinuous finite element equations, and convergence and error analysis.
For example, consider a nonlinear diffusion equation, which is one-dimensional: Firstly, define
, that is to say, the numerical solution we need is obtained by solving the differential equations.
Choosing a proper numerical flux is critical for the accuracy of DDG method.
is the maximum order of polynomials in two neighboring computing units.